Weak regularity of the inverse under minimal assumptions

Let $\Omega\subset\mathbb{R}^3$ be a domain and let $f\in BV_{\operatorname{loc}}(\Omega,\mathbb{R}^3)$ be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation $f^{-1}\in BV_{\operatorname{loc}}$. The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.